Those guidelines say that "active duty personnel may not engage in partisan political activities and all military personnel should avoid the inference that their political activities imply or appear to imply DoD sponsorship, approval, or endorsement of a political candidate, campaign, or cause."

US military authorities in the coalition did not immediately respond to CNN's request for comment.

Retired Rear Adm. John Kirby, a former Obama administration spokesperson and a CNN analyst, said on CNN's "The Situation Room" on Wednesday that service members having Trump sign the items was inappropriate.

"It is in fact a campaign slogan, that is a campaign item, and it is completely inappropriate for the troops to do this," Kirby said.

Kirby assigned some blame to Trump himself for political activity around the military: "Every time he's around military audiences, he tends to politicize it, and he brings in complaints and grievances from outside the realm of military policy."